逐像素贝塞尔曲线

如何解决逐像素贝塞尔曲线

Bresenham算法的一种变体适用于圆，椭圆和抛物线等二次函数，因此也应适用于二次Bezier曲线。

我打算尝试一种实现，但是随后我在网上找到了一个实现：http ://members.chello.at/~easyfilter/bresenham.html。

如果您需要更多细节或其他示例，上述页面上有指向100页PDF的链接，其中详细介绍了该方法：http ://members.chello.at/~easyfilter/Bresenham.pdf。

这是来自Alois Zingl网站的代码，用于绘制任何二次Bezier曲线。第一个例程将曲线在水平和垂直梯度变化处细分：

void plotQuadBezier(int x0, int y0, int x1, int y1, int x2, int y2)
{ /* plot any quadratic Bezier curve */
  int x = x0-x1, y = y0-y1;
  double t = x0-2*x1+x2, r;
  if ((long)x*(x2-x1) > 0) { /* horizontal cut at P4? */
    if ((long)y*(y2-y1) > 0) /* vertical cut at P6 too? */
      if (fabs((y0-2*y1+y2)/t*x) > abs(y)) { /* which first? */
        x0 = x2; x2 = x+x1; y0 = y2; y2 = y+y1; /* swap points */
      } /* Now horizontal cut at P4 comes first */
    t = (x0-x1)/t;
    r = (1-t)*((1-t)*y0+2.0*t*y1)+t*t*y2; /* By(t=P4) */
    t = (x0*x2-x1*x1)*t/(x0-x1); /* gradient dP4/dx=0 */
    x = floor(t+0.5); y = floor(r+0.5);
    r = (y1-y0)*(t-x0)/(x1-x0)+y0; /* intersect P3 | P0 P1 */
    plotQuadBezierSeg(x0,y0, x,floor(r+0.5), x,y);
    r = (y1-y2)*(t-x2)/(x1-x2)+y2; /* intersect P4 | P1 P2 */
    x0 = x1 = x; y0 = y; y1 = floor(r+0.5); /* P0 = P4, P1 = P8 */
  }
  if ((long)(y0-y1)*(y2-y1) > 0) { /* vertical cut at P6? */
    t = y0-2*y1+y2; t = (y0-y1)/t;
    r = (1-t)*((1-t)*x0+2.0*t*x1)+t*t*x2; /* Bx(t=P6) */
    t = (y0*y2-y1*y1)*t/(y0-y1); /* gradient dP6/dy=0 */
    x = floor(r+0.5); y = floor(t+0.5);
    r = (x1-x0)*(t-y0)/(y1-y0)+x0; /* intersect P6 | P0 P1 */
    plotQuadBezierSeg(x0,y0, floor(r+0.5),y, x,y);
    r = (x1-x2)*(t-y2)/(y1-y2)+x2; /* intersect P7 | P1 P2 */
    x0 = x; x1 = floor(r+0.5); y0 = y1 = y; /* P0 = P6, P1 = P7 */
  }
  plotQuadBezierSeg(x0,y0, x1,y1, x2,y2); /* remaining part */
}

第二个例程实际上绘制了一个Bezier曲线段（一个没有梯度变化的段）：

void plotQuadBezierSeg(int x0, int y0, int x1, int y1, int x2, int y2)
{ /* plot a limited quadratic Bezier segment */
  int sx = x2-x1, sy = y2-y1;
  long xx = x0-x1, yy = y0-y1, xy; /* relative values for checks */
  double dx, dy, err, cur = xx*sy-yy*sx; /* curvature */
  assert(xx*sx <= 0 && yy*sy <= 0); /* sign of gradient must not change */
  if (sx*(long)sx+sy*(long)sy > xx*xx+yy*yy) { /* begin with longer part */
    x2 = x0; x0 = sx+x1; y2 = y0; y0 = sy+y1; cur = -cur; /* swap P0 P2 */
  }
  if (cur != 0) { /* no straight line */
    xx += sx; xx *= sx = x0 < x2 ? 1 : -1; /* x step direction */
    yy += sy; yy *= sy = y0 < y2 ? 1 : -1; /* y step direction */
    xy = 2*xx*yy; xx *= xx; yy *= yy; /* differences 2nd degree */
    if (cur*sx*sy < 0) { /* negated curvature? */
      xx = -xx; yy = -yy; xy = -xy; cur = -cur;
    }
    dx = 4.0*sy*cur*(x1-x0)+xx-xy; /* differences 1st degree */
    dy = 4.0*sx*cur*(y0-y1)+yy-xy;
    xx += xx; yy += yy; err = dx+dy+xy; /* error 1st step */
    do {
      setPixel(x0,y0); /* plot curve */
      if (x0 == x2 && y0 == y2) return; /* last pixel -> curve finished */
      y1 = 2*err < dx; /* save value for test of y step */
      if (2*err > dy) { x0 += sx; dx -= xy; err += dy += yy; } /* x step */
      if ( y1 ) { y0 += sy; dy -= xy; err += dx += xx; } /* y step */
    } while (dy < 0 && dx > 0); /* gradient negates -> algorithm fails */
  }
  plotLine(x0,y0, x2,y2); /* plot remaining part to end */
}

该站点上还提供了抗锯齿代码。

来自Zingl站点的三次贝塞尔曲线的相应函数为

void plotCubicBezier(int x0, int y0, int x1, int y1,
  int x2, int y2, int x3, int y3)
{ /* plot any cubic Bezier curve */
  int n = 0, i = 0;
  long xc = x0+x1-x2-x3, xa = xc-4*(x1-x2);
  long xb = x0-x1-x2+x3, xd = xb+4*(x1+x2);
  long yc = y0+y1-y2-y3, ya = yc-4*(y1-y2);
  long yb = y0-y1-y2+y3, yd = yb+4*(y1+y2);
  float fx0 = x0, fx1, fx2, fx3, fy0 = y0, fy1, fy2, fy3;
  double t1 = xb*xb-xa*xc, t2, t[5];
  /* sub-divide curve at gradient sign changes */
  if (xa == 0) { /* horizontal */
    if (abs(xc) < 2*abs(xb)) t[n++] = xc/(2.0*xb); /* one change */
  } else if (t1 > 0.0) { /* two changes */
    t2 = sqrt(t1);
    t1 = (xb-t2)/xa; if (fabs(t1) < 1.0) t[n++] = t1;
    t1 = (xb+t2)/xa; if (fabs(t1) < 1.0) t[n++] = t1;
  }
  t1 = yb*yb-ya*yc;
  if (ya == 0) { /* vertical */
    if (abs(yc) < 2*abs(yb)) t[n++] = yc/(2.0*yb); /* one change */
  } else if (t1 > 0.0) { /* two changes */
    t2 = sqrt(t1);
    t1 = (yb-t2)/ya; if (fabs(t1) < 1.0) t[n++] = t1;
    t1 = (yb+t2)/ya; if (fabs(t1) < 1.0) t[n++] = t1;
  }
  for (i = 1; i < n; i++) /* bubble sort of 4 points */
    if ((t1 = t[i-1]) > t[i]) { t[i-1] = t[i]; t[i] = t1; i = 0; }
    t1 = -1.0; t[n] = 1.0; /* begin / end point */
    for (i = 0; i <= n; i++) { /* plot each segment separately */
    t2 = t[i]; /* sub-divide at t[i-1], t[i] */
    fx1 = (t1*(t1*xb-2*xc)-t2*(t1*(t1*xa-2*xb)+xc)+xd)/8-fx0;
    fy1 = (t1*(t1*yb-2*yc)-t2*(t1*(t1*ya-2*yb)+yc)+yd)/8-fy0;
    fx2 = (t2*(t2*xb-2*xc)-t1*(t2*(t2*xa-2*xb)+xc)+xd)/8-fx0;
    fy2 = (t2*(t2*yb-2*yc)-t1*(t2*(t2*ya-2*yb)+yc)+yd)/8-fy0;
    fx0 -= fx3 = (t2*(t2*(3*xb-t2*xa)-3*xc)+xd)/8;
    fy0 -= fy3 = (t2*(t2*(3*yb-t2*ya)-3*yc)+yd)/8;
    x3 = floor(fx3+0.5); y3 = floor(fy3+0.5); /* scale bounds to int */
    if (fx0 != 0.0) { fx1 *= fx0 = (x0-x3)/fx0; fx2 *= fx0; }
    if (fy0 != 0.0) { fy1 *= fy0 = (y0-y3)/fy0; fy2 *= fy0; }
    if (x0 != x3 || y0 != y3) /* segment t1 - t2 */
      plotCubicBezierSeg(x0,y0, x0+fx1,y0+fy1, x0+fx2,y0+fy2, x3,y3);
    x0 = x3; y0 = y3; fx0 = fx3; fy0 = fy3; t1 = t2;
  }
}

和

void plotCubicBezierSeg(int x0, int y0, float x1, float y1,
  float x2, float y2, int x3, int y3)
{ /* plot limited cubic Bezier segment */
  int f, fx, fy, leg = 1;
  int sx = x0 < x3 ? 1 : -1, sy = y0 < y3 ? 1 : -1; /* step direction */
  float xc = -fabs(x0+x1-x2-x3), xa = xc-4*sx*(x1-x2), xb = sx*(x0-x1-x2+x3);
  float yc = -fabs(y0+y1-y2-y3), ya = yc-4*sy*(y1-y2), yb = sy*(y0-y1-y2+y3);
  double ab, ac, bc, cb, xx, xy, yy, dx, dy, ex, *pxy, EP = 0.01;

  /* check for curve restrains */
  /* slope P0-P1 == P2-P3 and (P0-P3 == P1-P2 or no slope change) */
  assert((x1-x0)*(x2-x3) < EP && ((x3-x0)*(x1-x2) < EP || xb*xb < xa*xc+EP));
  assert((y1-y0)*(y2-y3) < EP && ((y3-y0)*(y1-y2) < EP || yb*yb < ya*yc+EP));
  if (xa == 0 && ya == 0) { /* quadratic Bezier */
    sx = floor((3*x1-x0+1)/2); sy = floor((3*y1-y0+1)/2); /* new midpoint */
    return plotQuadBezierSeg(x0,y0, sx,sy, x3,y3);
  }
  x1 = (x1-x0)*(x1-x0)+(y1-y0)*(y1-y0)+1; /* line lengths */
  x2 = (x2-x3)*(x2-x3)+(y2-y3)*(y2-y3)+1;
  do { /* loop over both ends */
    ab = xa*yb-xb*ya; ac = xa*yc-xc*ya; bc = xb*yc-xc*yb;
    ex = ab*(ab+ac-3*bc)+ac*ac; /* P0 part of self-intersection loop? */
    f = ex > 0 ? 1 : sqrt(1+1024/x1); /* calculate resolution */
    ab *= f; ac *= f; bc *= f; ex *= f*f; /* increase resolution */
    xy = 9*(ab+ac+bc)/8; cb = 8*(xa-ya);/* init differences of 1st degree */
    dx = 27*(8*ab*(yb*yb-ya*yc)+ex*(ya+2*yb+yc))/64-ya*ya*(xy-ya);
    dy = 27*(8*ab*(xb*xb-xa*xc)-ex*(xa+2*xb+xc))/64-xa*xa*(xy+xa);
    /* init differences of 2nd degree */
    xx = 3*(3*ab*(3*yb*yb-ya*ya-2*ya*yc)-ya*(3*ac*(ya+yb)+ya*cb))/4;
    yy = 3*(3*ab*(3*xb*xb-xa*xa-2*xa*xc)-xa*(3*ac*(xa+xb)+xa*cb))/4;
    xy = xa*ya*(6*ab+6*ac-3*bc+cb); ac = ya*ya; cb = xa*xa;
    xy = 3*(xy+9*f*(cb*yb*yc-xb*xc*ac)-18*xb*yb*ab)/8;
    if (ex < 0) { /* negate values if inside self-intersection loop */
      dx = -dx; dy = -dy; xx = -xx; yy = -yy; xy = -xy; ac = -ac; cb = -cb;
    } /* init differences of 3rd degree */
    ab = 6*ya*ac; ac = -6*xa*ac; bc = 6*ya*cb; cb = -6*xa*cb;
    dx += xy; ex = dx+dy; dy += xy; /* error of 1st step */
    for (pxy = &xy, fx = fy = f; x0 != x3 && y0 != y3; ) {
      setPixel(x0,y0); /* plot curve */
      do { /* move sub-steps of one pixel */
        if (dx > *pxy || dy < *pxy) goto exit; /* confusing values */
        y1 = 2*ex-dy; /* save value for test of y step */
        if (2*ex >= dx) { /* x sub-step */
          fx--; ex += dx += xx; dy += xy += ac; yy += bc; xx += ab;
        }
        if (y1 <= 0) { /* y sub-step */
          fy--; ex += dy += yy; dx += xy += bc; xx += ac; yy += cb;
        }
      } while (fx > 0 && fy > 0); /* pixel complete? */
      if (2*fx <= f) { x0 += sx; fx += f; } /* x step */
      if (2*fy <= f) { y0 += sy; fy += f; } /* y step */
      if (pxy == &xy && dx < 0 && dy > 0) pxy = &EP;/* pixel ahead valid */
    }
    exit: xx = x0; x0 = x3; x3 = xx; sx = -sx; xb = -xb; /* swap legs */
    yy = y0; y0 = y3; y3 = yy; sy = -sy; yb = -yb; x1 = x2;
  } while (leg--); /* try other end */
  plotLine(x0,y0, x3,y3); /* remaining part in case of cusp or crunode */
}

正如Mike’Pomax’Kamermans所指出的那样，该站​​点上三次Bezier曲线的解决方案并不完整。特别是三次贝塞尔曲线的抗锯齿问题，对有理三次贝塞尔曲线的讨论还不完整。

解决方法

我通过Google找到的二次/三次贝塞尔曲线代码主要是通过将线细分为一系列点并将它们与直线连接来工作的。光栅化发生在直线算法中，而不是贝塞尔曲线中。像布雷森汉姆（Bresenham）这样的算法可以逐像素地对一条线进行栅格化，并且可以对其进行优化（请参阅Po-
Han Lin的解决方案）。

什么是像线算法一样逐像素工作而不是通过绘制一系列点的二次贝塞尔曲线算法？